To reduce scan times or maximize temporal resolution, modern MRI protocols often perform undersampled acquisitions, particularly when using parallel imaging or acquiring dynamic data. Images are then estimated using advanced reconstruction techniques that leverage a priori information about the MRI system and/or signal. For example, the SENSE method for parallel image (PI) reconstruction requires knowledge of the sensitivity profiles of the phased-array receiver. Since this information is patient and system specific, it must be estimated using an empirical calibration procedure. Dynamic MRI approaches, such as kt-BLAST, often employ similar paradigms. Image reconstruction methods that require such information generally presume that this information is precise. Correspondingly, any errors in this information - e.g., from noise, numerical approximation, spatial misregistration, or physical effects like off-resonance - inevitably propagate into and compromise the final reconstructed images. Obtaining such information via a separate calibration scan or acquisition of an auto-calibration signal (ACS) also prolongs exam duration and can limit sampling freedom. To address these issues, there is emerging interest in the MRI community in developing training- and calibration-free image reconstruction methods, particularly for dynamic and parallel MRI. These techniques operate by structuring MRI data into specific matrix forms and promoting so-called low-rank (LR) structure, which is a higher-dimensional generalization of the sparsity concept that underlies compressed sensing. However, existing calibration-free PI methods construct and operate on very large matrices, resulting in extremely high computational and memory expenses. Training-free dynamic MRI methods commonly face limited degree-of-freedom (DOF) issues that result from the high anisotropy of the matrices constructed. We have recently developed an alternative strategy that operates by promoting low-rank structure locally rather than globally. In addition to circumventing the above DOF and computational limitations, our locally low-rank (LLR) reconstruction methods are naturally amenable to parallel implementation, and readily generalize for advanced applications like non-Cartesian and chemical shift encoded imaging. The objective of this proposal is the investigation and further development of LLR methods for MRI reconstruction. Aim 1 involves further mathematical development of the LLR strategy and of computational techniques that will enable its practical clinical application. Aim 2 focuses on generalizing LLR to a tensor model to more effectively handle MRI applications that acquire both parallel and dynamic (or parametric) data. The significance is that these methods can yield improved reconstructions on undersampled higher- dimensional MRI acquisitions, are inherently generic, and do not require training or calibration procedures. Thus, virtually all of current clinical MRI could benefit from the improved resolution, signal quality, and/or reduced acquisition times that these techniques will facilitate or the novel applications they may enable.